When Nine Points Are Worth But Eight: Euler’s Resolution of Cramer’s Paradox - Intersection of Lines and Curves

Two straight lines in the plane intersect in one point. This statement is always true, except when the two lines are parallel (although the statement is always true in the projective plane). And like the previous question, the problem of intersection can be considered for curves of higher degree. For example, a line can typically intersect an ellipse, parabola or hyperbola in two points. Two conics can intersect in as many as 4 points. By considering various low order cases, many 18th century mathematicians came to the conclusion that two curves of order \(m\) and \(n\) typically intersect in \(mn\) points, and can never intersect in more than \(mn\) points. This result is now called Bézout's Theorem, after Etienne Bézout (1730-1783), who gave the first acceptable proof of this result in 1779.

Figure 1.  A polynomial of degree three and an ellipse may intersect in six points. (Image created using GeoGebra.)

Hold on, now … there's a problem here! Let's suppose \(m=n=3.\) Then on the one hand, two curves of order three will typically intersect in 9 points. On the other hand, if \(n=3,\) then \(\frac{n^2+3n}{2}=9,\) so those same 9 points should have been enough to determine a unique curve of order three! This apparent contradiction goes by the name of Cramer's Paradox, although it was first noticed by Maclaurin and it was resolved at least as successfully by Euler as it was by Cramer.

In this article, we will look at the problem of fitting points to curves and use some modern notions from linear algebra to understand how Cramer's Paradox is resolved. We will also look at the work of Euler and Cramer, who grappled with this problem but didn't have the benefit of such notions as linear independence and rank in order to guide their thoughts. We will also get a glimpse of a letter that Euler wrote to Cramer in 1744, which had been unknown until 2003 but will soon appear in Euler's collected works, in which Euler first suggested a resolution for Cramer's Paradox.